I like this question because it asks us to get at the heart of how kids (and adults untainted by math instruction) conceptualize numbers. I responded that I thought 1 1/2 made more sense because 3 sixths would require an implicit cutting of all of the pieces of what I interpreted to be the quesadilla (because, really, who splits a cookie into thirds? Who shares cookies at all?) into six pieces, in order to share them equally. I applied what I thought Barbara and Sebastian would do, which is to take one piece each, then split the remaining piece in half.
I asked Barbara and Sebastian this question, using a sheet of paper I'd cut into thirds, and presented it like this:
I'm going to take this paper and cut it into three equal-sized pieces, and then you and I are each going to take the same amount of paper. This is called 1/3, this is called 1/3, and this is called 1/3.
Barbara gave herself two of the pieces and gave me one. She was done.
I made a face and told her this wasn't fair. She switched so I had 2/3 and she had 1/3. I made another face and she said, "or, we could rip this one." She ripped the third piece, so now we each had one of the original 1/3 pieces and half of the remaining 1/3 piece. I asked her how much she had and she said I have one piece, and a smaller piece.
I asked her what the smaller piece was called and she was stumped. "A little piece." I reported that this was the answer our kids gave, committing the sin of assuming understanding based on the most advanced student's response.
I tried the same thing with Sebastian today. Though there are some highlights, my main takeaway was that I wanted to have a conversation about math, making it relevant by using the fact that he'd just asked me for a peanut butter sandwich, but he really just wanted a peanut butter sandwich. When talking math with kids, timing is everything:
1. Both of our kids first thought that the best way to share was to take 2/3 for themselves and leave me 1/3. This is either an indictment or a triumph of our parenting style.
2. This moment with Sebastian, feeling silly and hungry, reminded me that often, when we think we're talking math with our kids, they are really more into the surface structure of the task. The underlying work with number sense, mathematical problem solving, and struggle to name fractional amounts take up no more than 25% of Sebastian's working memory, the rest of it filled with 5-year-old versions of ideas like "why can't you just make your own damn sandwich?" and "now can I eat it?"
3. After thinking about this for the last two days, and trying this with both Barbara and Sebastian, I am struck by the ease with which each of them decided to split the remaining piece, and the lack of vocabulary to describe the precise amount that each piece represented. I was also struck that, by essentially telling them that each of the three pieces was 1/3 of the whole, I was naming a new unit for them, giving a name to what they otherwise would call a "piece." I don't know enough about early childhood development to describe this precisely, but they are at a stage where they count things. I could have called each third a whatsit, and they would have described the final amount the same way: A whatsit and a smaller piece. The term 1/3 means nothing as of right now, but the concept of splitting things into equal-sized parts makes a lot of sense. Not in some abstract sense, but in a very real, rip-the-bread kind of way.
4. Another take-away for me was that the reason we wouldn't accept either kid's answer beyond a certain age is that, at least in math classes, we expect every part of an amount to be expressed in terms of a single unit. If the unit is the whole sandwich (or quesadilla, or piece of paper), we want to know how much of it we have. If the unit is a third, we want to know how many thirds. We don't want to know how many thirds and how many other pieces, nor do we ask for the number of thirds and fourths. Egyptians didn't write fractions this way back in the day, expressing fractions as the sum of unit fractions, but I'm not taking this as evidence that they actually perceived fractions that way. I have a hard time believing that the pharaohs really conceived of Sebastian's original partition (2/3) as 1/3 + 1/4 + 1/12, meaning "OK, let's take one of these original pieces, mark it for later, put the original back together and cut into four pieces, take one of those and mark it, then put the original back together and cut into twelve pieces, mark it, and take the marked pieces. No, wait; that's not even. I don't know how they viewed fractions conceptually, but I know that our current representation differs from how both of my kids (small sample size, grew up in the same home, but still) view fractions, and I imagine this concept/notation gap is not unique to our household or culture.
The same-unit idea is a new one for me, though. It got me to thinking that we don't say things like "the pencil was five inches and three centimeters long", but we can say, "the table was three feet, six inches tall" because inches and feet share a common unit (the unit either being the inch, decomposed from feet, or the foot, composed from inches.) For what it's worth, I think "three and a half feet" very clearly refers to the foot as the unit, whereas "three feet, six inches" uses the inch as the basic unit. But at some point, kids realize that an answer to "how much of the quesadilla do you have?" should not be answered with "one piece of this size, and one piece of this unrelated size."
5. Perhaps most importantly, I need to spend more time talking math with my kids. Not because they need to get ahead, or because of increasing global competition, but because it lets the people I love the most fall in love with the subject I love the most, and makes us all think in the process.